Elements of Integral Equations

Integral equations appear everywhere in applied mathematics, physics, and engineering. This book provides a structured introduction that takes you from definitions to the core methods without skipping the proofs.

The first chapter sets up the classification: Fredholm and Volterra equations of the first, second, and third kind. Singular equations, convolution-type equations, eigenvalues and eigenfunctions, and the Leibniz rule for differentiation under the integral sign.

Chapter two covers the functional analysis foundation: square integrable functions, inner products, orthogonality, norms, and the Cauchy-Schwarz inequality. The trial method for verifying solutions is presented with worked examples, along with a preview of the Neumann series.

The core of the book handles the bidirectional conversion between differential and integral equations. Initial value problems and boundary value problems are both covered, with the Green’s function connection made explicit.

Topics Covered

• Fredholm and Volterra equations (all three kinds)
• Kernels: symmetric, separable, singular, and convolution-type
• Square integrable functions and norms
• Converting ODEs to integral equations and back
• Green’s functions and their connection to integral equations
• Existence and uniqueness theorems
• Fredholm theory overview